-
UNIVERSITY OF CALIFORNIA
Santa Barbara
Travelling-Wave Photodetectors
by
Kirk Steven Giboney
A dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
in
Electrical and Computer Engineering
Committee in Charge
Professor John E. Bowers, Co-Chairperson
Professor Mark J. W. Rodwell, Co-Chairperson
Professor Larry A. Coldren
Professor Umesh K. Mishra
August 1995
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The dissertation of Kirk Steven Giboney is approved
August 1995
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August 4, 1995
Travelling-Wave Photodetectors
Copyright © 1995
by Kirk Steven Giboney
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ACKNOWLEDGEMENTS
– Beauty makes our pursuits worthwhile. –
Progress flows only from collective efforts. Individuals can
claim no more than
incremental contributions. Even so, the aptly exploited talents
are not earned, but
merely developed. Like our talents, our associations with our
contemporaries are gifts
that we choose to develop. I would like to thank several people
who contributed to
my education and this work.
A thesis advisor is a lot like a parent for the graduate
student's intellectual and
professional development. My advisors, Mark Rodwell and John
Bowers, took their
roles as seriously. Their technical guidance, support, and
impetus, enabled this work,
and greatly enhanced my life. My committee, which also included
Larry Coldren and
Umesh Mishra, made whole-hearted efforts not only to direct
outstanding research,
but to communicate the personal and professional aspects crucial
to success.
The faculty have set an invaluable example of cooperation and
collaboration that
has been enthusiastically embraced throughout the department by
staff, graduate
students, and visiting and postdoctoral researchers. Radha
Nagarajan helped me
develop the process I used. Tom Reynolds balanced a difficult
job of keeping the labs
running while making significant contributions to individual
research, such as the
anti-reflection coatings on the travelling-wave photodetectors.
Rich Mirin made sure
the material he grew was appropriate and done right the first
time. I learned much
about high-speed photodetectors from Yih-Guei Wey while doing
electro-optic
sampling measurements on his.
Daily interactions among researchers contribute substantially to
research results,
professional maturity, and personal satisfaction. Many fruits of
my interactions with
Scott Allen and Masayuki Kamegawa, particularly related to
device processing, were
incorporated into this work. Judy Karin helped me get started on
the optical bench.
Dan Tauber, Ralph Spickerman, and Mike Case were comrades in
microwave devices
and slow-wave effects, and together, we planted many seeds of
ideas. Dennis
Derickson showed me the detailed workings of mode-locked
semiconductor lasers. I
worked with Mark Mondry at McDonnell Douglas before we came to
UCSB, and we
engaged in many discussions on a wide range of subjects at both
places. Anish Goyal
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and Dubravko Babic were always willing to discuss anything and
offer help. Dan
Cohen supported Professor Coldren's group and managed to help
others while
working on his thesis.
It has been a pleasure to work alongside many others. In
Professor Bowers'
group were Roger Helkey, Wenbin Jiang, Gary Wang, Chi-Kuang Sun,
Rajeev
Ram, Jim Dudley, Alan Mar, Pat Corvini, John Wasserbauer, Debbie
Crawford,
Paul Morton, Peter Blixt, Anders Petersen, Aaron Hawkins, and
Kehl Sink. In
Professor Rodwell's group were Ruai Yu, Eric Carman, Kimi Abe,
Yoshiyuki
Konishi, Uddalak Bhattacharya, Madhukar Reddy, Bipul Agarwal,
Rajasekhar
Pullela, and James Guthrie. There were also Scott Campbell,
Joyce Olsen, Jong
Chang Yi, Chi-Ping Chao, Eva Strzelecka, and many others. My
special appreciation
goes to Lorene Inouye for helping me to keep in balance.
I cherish our friendships and experiences together far above any
other aspect of
my time at UCSB. I am happy that you all shared your time and
efforts with me.
I thank ARPA Optoelectronics Technology Center and Ultra
Program
administered by Anis Husain and Robert Leheny, Charles
Tsacoyeanes at Rome
Laboratories, Joe Weller at the ONT Block Program on
Electro-Optics Technology,
and the United States Congress for their support. I am grateful
to the American
taxpayers for providing funding for this work, and to the
California taxpayers for
funding the University of California.
I especially appreciate my parents and grandmother, and my
brother and sister
and their families for their support and love, and for helping
me to keep my values
straight.
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VITAJuly 19, 1960 Born, Bakersfield, California, USA
December 1984 B.S. Physics, University of California, Davis
August 1985 – August 1988 Senior Engineer, McDonnell Douglas
Astronautics
Company, Huntington Beach, California
December 1990 M.S. Electrical and Computer Engineering,
University
of California, Santa Barbara
April 1989 – July 1995 Graduate Student Researcher, University
of California,
Santa Barbara
July 1990 – July 1995 Technical Consultant, Hughes Research
Laboratories,
Malibu, California
July 1995 – Research Intern VI, Hewlett-Packard
Laboratories,
Palo Alto, California
Publications
34. Yih-Guei Wey, Kirk Giboney, John Bowers, Mark Rodwell,
Pierre Silvestre, PrabhuThiagarajan, and Gary Robinson, "110 GHz
GaInAs/InP Double Heterostructure p-i-nPhotodetectors," J.
Lightwave Technol., vol. 13, no. 7, pp. 1490-1499, Jul., 1995.
33. Kirk Giboney, John Bowers, and Mark Rodwell,
"Travelling-Wave Photodetectors," in 1995IEEE MTT-S Microwave
Symposium Digest, pp. 159-162, May 15-19, 1995. (Invited)
32. Kirk S. Giboney, Mark J. W. Rodwell, and John E. Bowers,
"Field-Screening Effects inTravelling-Wave and Vertically
Illuminated p-i-n Photodetectors," presented at 1995 Conf.Lasers
Electro-Optics, Baltimore, MD, May, 1995, CFB4.
31. Kirk S. Giboney, Radhakrishnan L. Nagarajan, Thomas E.
Reynolds, Scott T. Allen,Richard P. Mirin, Mark J. W. Rodwell, and
John E. Bowers, "Travelling-WavePhotodetectors with 172 GHz
Bandwidth and 76 GHz Bandwidth-Efficiency Product," IEEEPhoton.
Technol. Lett., vol. 7, no. 4, pp. 412-414, Apr., 1995.
30. Kirk S. Giboney, Scott T. Allen, Mark J. W. Rodwell, and
John E. Bowers, "PicosecondMeasurements by Free-Running
Electro-Optic Sampling," IEEE Photon. Technol. Lett.,vol. 6, no.
11, pp. 1353-1355, Nov., 1994.
29. Radhakrishnan Nagarajan, Kirk Giboney, Rangchen Yu, Daniel
Tauber, John Bowers, andMark Rodwell, "High-Speed Optoelectronics,"
presented at 21st Int. Symp. CompoundSemicond., San Diego, CA,
Sep., 1994, WP3.1. (Invited)
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28. R. H. Walden, W. E. Stanchina, R. A. Metzger, R. Y. Loo, J.
Schaffner, M. W. Pierce, Y.K. Brown, F. Williams, V. Jones, J.
Pikulski, M. Rodwell, K. Giboney, R. A. Mullen, R.WongQuen, and J.
F. Jensen, "InP-Based Optoelectronic Integrated Receiver
Front-EndsUsing Heterojunction Bipolar Transistors and
Base-Collector Photodiodes," presented atEng. Found. Third Conf.
High Speed Optoelectron. Devices for Commun. andInterconnects,
Shell Beach, CA, Aug., 1994. (Invited)
27. Kirk Giboney, Mark Rodwell, and John Bowers,
"Travelling-Wave Photodetectors,"presented at Eng. Found. Third
Conf. High Speed Optoelectron. Devices for Commun.
andInterconnects, Shell Beach, CA, Aug., 1994. (Invited)
26. Kirk Giboney, Radhakrishnan Nagarajan, Thomas Reynolds,
Scott Allen, Richard Mirin,Mark Rodwell, and John Bowers, "172 GHz,
42% Quantum Efficiency p-i-n Travelling-Wave Photodetector,"
presented at 52nd Annual Device Res. Conf., Boulder, CO, Jun.,1994,
VIA-9.
25. Kirk S. Giboney, Scott T. Allen, Mark J. W. Rodwell, and
John E. Bowers, "1.5 ps Fall-Time Measurements by Free-Running
Electro-Optic Sampling," presented at Conf. LasersElectro-Optics,
Anaheim, CA, May, 1994, CME4.
24. R. H. Walden, W. E. Stanchina, R. A. Metzger, R. Y. Loo, J.
Schaffner, M. W. Pierce, Y.K. Brown, F. Williams, V. Jones, J.
Pikulski, M. Rodwell, K. Giboney, R. A. Mullen,and J. F. Jensen,
"Broadband optoelectronic integrated receiver front-ends comprising
InP-based heterojunction bipolar transistors and base-collector
photodiodes," presented at Conf.Optical Fiber Commun., San Jose,
CA, Feb., 1994.
23. Yih-Guei Wey, Kirk S. Giboney, John E. Bowers, Mark J. W.
Rodwell, Pierre Silvestre,Prabhu Thiagarajan, and Gary Y. Robinson,
"108 GHz GaInAs/InP p–i–n Photodiodes withIntegrated Bias Tees and
Matched Resistors," IEEE Photon. Technol. Lett., vol. 5, no. 11,pp.
1310-1312, Nov., 1993.
22. John Bowers, Kirk Giboney, Yih-Guei Wey, Mark Rodwell, "New
Concepts in 100 GHzHigh-Efficiency Photodetectors," presented at
LEOS Summer Topical Meeting OpticalMicrowave Interactions, Santa
Barbara, Jul., 1993, M1.1. (Invited)
21. Kirk S. Giboney, Yih-Guei Wey, John E. Bowers, Mark J. W.
Rodwell, Pierre Silvestre,Prabhu Thiagarajan, and Gary Y. Robinson,
"High-Speed GaInAs/InP p-i-n Photodiodeswith Integrated Bias Tees,"
presented at Fifth Int. Conf. Indium Phosphide and RelatedMater.,
Paris, France, Apr., 1993, TuE5.
20. Yih-Guei Wey, Kirk S. Giboney, John E. Bowers, Mark J. W.
Rodwell, Pierre Silvestre,Prabhu Thiagarajan, and Gary Y. Robinson,
"110 GHz Double Heterostructure GaInAs/InPp-i-n Photodiode," in OSA
Proc. Ultrafast Electron. Optoelectron., vol. 14, pp.
45-48,1993.
19. Kirk S. Giboney, Mark J. W. Rodwell, and John E. Bowers,
"Traveling-WavePhotodetectors," IEEE Photon. Technol. Lett., vol.
4, no. 12, pp. 1363-1365, Dec., 1992.
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18. J. G. Wasserbauer, D. J. Derickson, K. Giboney, R. J.
Helkey, J. R. Karin, A. Mar, and J.E. Bowers, "Integrated Optical
Transmitters and Receivers Using Multi-Segment LaserProcesses,"
presented at LEOS Summer Topical Meeting Optical Microwave
Interactions,Santa Barbara, Jul., 1992.
17. Eric Carman, Michael Case, Masayuki Kamegawa, Ruai Yu, Kirk
Giboney, and M. J. W.Rodwell, "V-Band and W-Band Broad-Band,
Monolithic Distributed Frequency Multipliers,"IEEE Microwave Guided
Wave Lett., vol. 2, no. 6, pp. 253-254, Jun., 1992.
16. Eric Carman, Michael Case, Masayuki Kamegawa, Ruai Yu, Kirk
Giboney, and M. J. W.Rodwell, "V-Band and W-Band Broad-Band,
Monolithic Distributed Frequency Multipliers,"presented at 1992
IEEE MTT-S Int. Microwave Symp., Albuquerque, NM, Jun., 1992
(in1992 IEEE MTT-S Microwave Symposium Digest, pp. 819-22, Jun.
1-5, 1992).
15. Eric Carman, Michael Case, Masayuki Kamegawa, Ruai Yu, Kirk
Giboney, and M. J. W.Rodwell, "Electrical Soliton Devices as
>100 GHz Signal Sources," presented at UltrafastPhenomena VIII
Conf., Antibes, France, Jun., 1992.
14. M. J. W. Rodwell, Scott Allen, Masayuki Kamegawa, Kirk
Giboney, Judy Karin, MichaelCase, Ruai Y. Yu, and J. E. Bowers,
"Picosecond Photodetectors Monolithically Integratedwith High-Speed
Sampling Circuits," presented at AFCEA DOD Fiber Opt. Conf.,
Mar.,1992.
13. Mark J. W. Rodwell, Masayuki Kamegawa, Ruai Yu, Michael
Case, Eric Carman, and KirkS. Giboney, "GaAs Nonlinear Transmission
Lines for Picosecond Pulse Generation andMillimeter-Wave Sampling,"
IEEE Trans. Microwave Theory Tech., vol. 39, no. 7, pp.1194-1204,
Jul., 1991.
12. M. Kamegawa, K. Giboney, J. Karin, S. Allen, M. Case, R. Yu,
M. J. W. Rodwell, and J.E. Bowers, "Picosecond GaAs Monolithic
Optoelectronic Sampling Circuit," IEEE Photon.Technol. Lett., vol.
3, no. 6, pp. 567-569, Jun., 1991.
11. Y. G. Wey, D. L. Crawford, K. Giboney, J. E. Bowers, M. J.
Rodwell, P. M. Sylvestre,M. J. Hafich, and G. Y. Robinson,
"Ultrafast graded double-heterostructure GaInAs/InPphotodiode,"
Appl. Phys. Lett., vol. 58, no. 19, pp. 2156-2158, May 13,
1991.
10. M. J. W. Rodwell, Masayuki Kamegawa, Michael Case, Ruai Y.
Yu, and Kirk Giboney,Eric Carman, Judy Karin, Scott Allen, and Jeff
Franklin, "Nonlinear Transmission Linesand their Applications in
Picosecond Optoelectronic and Electronic Measurements,"presented at
Eng. Found. Conf. High Freq./High Speed Optoelectron., Palm Beach,
FL,Mar., 1991.
9. Y. Wey, D. Crawford, K. Giboney, A. Mar, J. Bowers, "Graded
Double HeterostructurePhotodetectors," presented at Eng. Found.
Conf. High Freq./High Speed Optoelectron.,Palm Beach, FL, Mar.,
1991.
8. D. L. Crawford, Y. G. Wey, K. Giboney, M. Rodwell, J. Bowers,
P. Sylvestre, M. Hafich,and G. Robinson, "3.8 ps FWHM Impulse
Response of a Graded Double Heterostructure P-I-N Photodiode
Fabricated on a Semi-Insulating Substrate," presented at Third Int.
Conf.Indium Phosphide and Related Mater., Cardiff, Wales, 1991.
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7. D. L. Crawford, Y. G. Wey, J. E. Bowers, K. Giboney, and M.
Rodwell, "New Directionsin High Speed Photodetectors," presented at
Conf. Lasers Electro-Optics, Baltimore, MD,1991, CWB6.
(Invited)
6. Michael Case, Eric Carman, Masayuki Kamegawa, Kirk Giboney,
Ruai Yu, Kathryn Abe,M. J. W. Rodwell, and Jeff Franklin, "Impulse
Generation and Frequency MultiplicationUsing Soliton Effects in
Monolithic GaAs Circuits," in OSA Proc. Picosecond
Electron.Optoelectron., vol. 9, pp. 140-144, 1991.
5. Masayuki Kamegawa, K. Giboney, J. Karin, S. Allen, M. Case,
R. Yu, M. J. W. Rodwell,and J. E. Bowers, "Picosecond GaAs
Photodetector Monolithically Integrated with a High-Speed Sampling
Circuit," in OSA Proc. Picosecond Electron. Optoelectron., vol. 9,
pp.104-107, 1991.
4. D. L. Crawford, Y. G. Wey, K. Giboney, J. E. Bowers, M. J.
Rodwell, P. M. Sylvestre,M. J. Hafich, and G. Y. Robinson,
"Ultrafast Graded Double-Heterostructure p-i-nPhotodiode," in OSA
Proc. Picosecond Electron. Optoelectron., vol. 9, pp. 92-96,
1991.(Invited)
3. Eric Carman, Kirk Giboney, Michael Case, Masayuki Kamegawa,
Ruai Yu, Kathryn Abe,M. J. W. Rodwell, and Jeff Franklin, "28–39
GHz Distributed Harmonic Generation on aSoliton Nonlinear
Transmission Line," IEEE Microwave Guided Wave Lett., vol. 1, no.
2,pp. 28-31, Feb., 1991.
2. Y. G. Wey, M. Kamegawa, A. Mar, K. J. Williams, K. Giboney,
D. L. Crawford, J. E.Bowers, and M. Rodwell, "Hybrid Integration of
an InGaAs/InP PIN Photodiode with anUltrafast Sampling Circuit,"
presented at Conf. Optical Fiber Commun., San Diego, CA,1991,
PD8-1.
1. M. Case, M. Kamegawa, K. Giboney, M. Rodwell, J. Franklin,
and J. E. Bowers, "62.5 psto 5.5 ps Soliton Compression on a
Monolithic Nonlinear Transmission Line," presented at48th Annual
Device Res. Conf., Santa Barbara, CA, Jun., 1990, VA-2.
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ABSTRACT
Travelling-Wave Photodetectors
by
Kirk Steven Giboney
Photodetector efficiency decreases as bandwidth increases.
Bandwidth-efficiencyproducts of vertically illuminated
photodetectors are limited to about 40 GHz. Thisproduct imposes a
bound on the speed and sensitivity of photoreceivers used inoptical
transmission systems.
Waveguide photodetectors are an attractive option for increasing
the bandwidth-efficiency product over the intrinsic limit of
vertically illuminated photodetectors. Byguiding the illumination
perpendicular to the carrier drift field, the inherent
tradeoffbetween efficiency and the transit-time bandwidth
limitation is diminished. However,even higher bandwidth-efficiency
products are possible with consideration for thepropagation of the
electrical waves to the load.
Attention to the microwave design of waveguide photodetectors
leads totravelling-wave photodetectors. A travelling-wave
photodetector is a waveguidephotodetector with an electrode
structure designed to support travelling electricalwaves with
characteristic impedance matched to that of the external circuit.
Thetravelling-wave photodetector is thus modelled by a matched
section of transmissionline with an exponentially decaying
photocurrent source propagating on it at theoptical group
velocity.
The mismatch between the group velocity of the photocurrent
source and thephase velocities of the electrical waves it generates
limits travelling-wavephotodetector bandwidth. The
velocity-mismatch bandwidth limitation is essentiallyindependent of
device length, so a travelling-wave photodetector can arbitrarily
bemade long enough for nearly 100% internal quantum efficiency
withoutcompromising bandwidth. A simple form for the
velocity-mismatch bandwidthlimitation is derived that affords
physical insight and provides a basis for usingtraditional design
methods.
The first theory, fabrication, and measurement of
travelling-wave photodetectorscomprise this thesis work. New
developments in electro-optic techniques enablemeasurements of
bandwidths as high as 190 GHz, the highest reported for a
p-i-nphotodetector by more than 70%. The travelling-wave
photodetectors displaybandwidth-efficiency products as large as 84
GHz, breaking the record for anyphotodetector without gain by 50%.
Comparisons with vertically illuminated andwaveguide photodetectors
fabricated on the same wafer establish the advantage
oftravelling-wave photodetectors.
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CONTENTS
1 INTRODUCTION ... . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 11.1 Background 1
1.2 Range and Organization of this Dissertation 8
References 10
2 THEORY OF DISTRIBUTED PHOTODETECTION... . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 132.1
Photodetector-Electrical Waveguide 14
2.2 Equivalent-Circuit Model 20
2.3 Velocity-Mismatch Impulse Response 28
2.4 Velocity-Mismatch Bandwidth Limitation 34
2.5 Space-Charge Field Screening 40
References 44
3 DESIGN & FABRICATION ... . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 473.1 Hybrid-Coplanar Travelling-Wave Photodetector
47
3.2 Bandwidth Model 51
3.3 Device Design 53
3.4 Fabrication 60
References 70
4 ELECTRO-OPTIC MEASUREMENT
TECHNIQUES.................................. 734.1 Electro-Optic
Probe Station 73
4.2 Electro-Optic Sampling 78
4.3 System Response 83
4.4 Deconvolution 86
4.5 Noise & Interference 89
References 95
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5 MEASUREMENTS & ANALYSIS .. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 995.1 Travelling-Wave Photodetector Performance 99
5.2 Propagation Constant 105
5.3 Electro-Optic Sampling System Response Correction 111
5.4 Reflection Deconvolution 115
5.5 Field-Screening Effects 117
References 122
6 OVERVIEW... . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 1236.1 Summary 123
6.2 Further Analysis & Improvements 125
6.3 Periodic Travelling-Wave Photodetector 127
6.4 Applications 131
References 132
A TRANSVERSE-RESONANCE SOLUTION
PROGRAM............................... 135A.1 Main Script 135
A.2 Specific Functions 140
A.3 General Functions & Scripts 142
References 143
B FABRICATION PROCESS .. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 145B.1 Process Plan 145
B.2 Cleaning, Photolithography, & Spin-On Films 154
B.3 Chemical Specifications & Preparation 157
References 158
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CHAPTER 1
INTRODUCTION
Optical fiber communications is the primary means of
transmitting information
over long distances today. The National Information
Infrastructure, or "Information
Superhighway," envisions a large-scale, high-speed computer
network covering the
entire nation and connecting with the rest of the world. Full
realization of this plan
will require large increases in the transmission capacity of our
optical communications
systems. This will be achieved by increasing transmission rates
through greater time-
division multiplexing (TDM), and further exploiting the optical
bandwidth of silica
fibers with optical-frequency-division multiplexing (OFDM).
Optoelectronic instrumentation must keep a step ahead of the
communications
industry and also keep pace with developments in short-pulse
lasers. Commercially
available mode-locked solid-state lasers that produce ultrafast
optical pulses are
driving new scientific, medical, manufacturing, and military
applications.
Conventional photodetectors and electronics that have routinely
met the demands of
optoelectronic systems are now being pressed to their limits and
beyond.
This work encompasses the invention of the travelling-wave
photodetector
(TWPD). With this new device, the largest simultaneous bandwidth
and efficiency of
a photodetector is demonstrated, and a significantly higher
fundamental limit on
photodetector performance is exposed. The introduction of the
TWPD contributes to
continuing advances in high-speed optoelectronics and will
enable larger capacity in
future optoelectronic systems.
1 . 1 Background
A photodetector is an optical-to-electrical transducer.
High-speed photodetectors
are capable of converting high-modulation-frequency optical
signals into electrical
signals. Efficient photodetectors convert a large fraction of
the light to electricity.
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2 1. Introduction
C R I
Fig. 1.1 Simplified circuit diagram of a lumped-element
photodetector and load showing thephotocurrent source, capacitance
of the depletion region, and load resistance.
High-speed photodetectors must have good efficiency to be
useful, so bandwidth and
efficiency are the primary measures of high-speed photodetector
performance.
A photodetector can be viewed as a two port device with an
optical input and an
electrical output. As a linear system, it is completely
characterized by its impulse or
frequency response. Photodetector performance is conventionally
summarized by
specifying two parameters: bandwidth and quantum efficiency.
Bandwidth is
generally taken to be the lowest frequency at which the
magnitude electrical response
is less than -3 dB from the DC (zero frequency) response.
Quantum efficiency is also
called external quantum efficiency, the ratio of unit charges
collected to photons
incident. The bandwidth-efficiency product is an important
figure of merit for
photodetectors.
A simplified circuit diagram of a lumped-element photodetector
is drawn in Fig.
1.1. The bandwidth of such a device is limited by the response
of the current source
and the response of the overall circuit. The response of the
current source, transit
response, applies to all photodetectors and is determined by the
velocities of
photogenerated electrons and holes and the distances they
travel. Assuming carriers
are uniformly generated across the depletion layer and the
carrier velocities are
constant and equal, the transit bandwidth limitation is given
by
Bt = 0.55
vdd
(1.1)
where vd is the electron and hole drift velocity, and d is the
thickness of the depletion
region, which is the distance travelled by the carriers.
Clearly, large bandwidth
requires a thin depletion region.
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1.1 Background 3
p
i
n
Absorber R E
Light
Fig. 1.2 Schematic diagram of a vertically illuminated
photodetector.
The circuit RC response applies only to lumped-element
photodetectors, and it
depends on the area and thickness of the depletion region and
the series resistance.
The RC bandwidth limitation is given by
BRC =
12πRC
(1.2)
where R is the series resistance. C = εA d, where ε is the
dielectric constant of thedepletion layer and A is its area.
The general form for external quantum efficiency of a
photodetector is
η = ηc 1− R( ) 1− e−Γαl( ) (1.3)
where ηc is the modal coupling efficiency, R is the Fresnel
reflectivity, Γ is the
modal confinement factor, α is the optical power absorption
coefficient of theabsorbing layer material, and l is the optical
absorption path length in the depletion
region.
The vertically illuminated photodetector (VPD), drawn
conceptually in Fig. 1.2,
is the most common type. Light is incident perpendicular to the
device layers in this
type of device. All of the light is coupled directly into the
absorbing layer in a VPD,so
ηc = 1 and Γ = 1, and if light passes through only once, l = d.
A thin depletion
layer is required for large bandwidth, according to (1.1),
however this reduces the
quantum efficiency in (1.3). In the limit of a thin depletion
layer, (1.3) becomes
η ≈ 1− R( )αd. A fundamental limit to the bandwidth-efficiency
product for VPDs
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4 1. Introduction
can be derived from (1.1) and (1.3) by assuming that the
junction area, A, can be
decreased arbitrarily to eliminate the RC bandwidth limitation
imposed in (1.2). The
bandwidth-efficiency product then takes on a simple form for
VPDs
Bη = 0.55αvd (1.4)
where the reflectivity has been set to zero [1]. High-field
electron and hole velocities
in GaAs are about 70 nm/ps. Assuming α = 1 µm-1, the maximum
possiblebandwidth-efficiency product for a single-pass GaAs VPD is
about 40 GHz.
In light of (1.4), an obvious means of obtaining high
bandwidth-efficiency
product is to choose a material that has a large product of
carrier velocity and
absorption coefficient at the wavelength range of interest.
Availability of materials and
fabrication resources severely restricts the prospects of
homogeneous, or bulk,
semiconductors, however quantum confinement structures offer
additional
possibilities.
Quantum wells, wires, or dots (one, two, or three dimensional
quantum
confinement structures) have peaks in the absorption coefficient
over narrow bands of
photon energies. The carrier mobility and saturated drift
velocity is also enhanced in
quantum wells and wires due to suppression of scattering
mechanisms. A theoretical
study of quantum wire photodetectors concluded that the increase
in absorption in the
wires is likely to be compromised by dimensional and
compositional variations, and
further offset by the fill-factor [2]. Large drift velocity
enhancements over saturated
bulk values were assumed to predict bandwidth-efficiency product
improvements
over bulk material. The temperature dependence of the absorption
spectra in quantum
structures can make temperature stabilization necessary.
There are several approaches to increasing the
bandwidth-efficiency product over
the inherent VPD limit expressed in (1.4). Mirrors are used to
reflect light through the
absorber several times in the resonant-cavity-enhanced (RCE)
photodetector, as
depicted in Fig. 1.3 [3-4]. This effectively increases the
absorption length, l in (1.3),
but not the distance travelled by the carriers, d in (1.1). The
efficiency can then go to
unity while the bandwidth is still determined by (1.1) and
(1.2). Added series
resistance due to the mirrors has limited p-i-n RCE bandwidths
to 17 GHz [5],
although this has apparently been overcome in
metal-semiconductor-metal (MSM)
-
1.1 Background 5
p
i
n
Absorber R E
Light
Mirrors
Fig. 1.3 Schematic diagram of a resonant-cavity-enhanced
photodetector.
type devices showing bandwidths over 40 GHz [6]. The absorption
spectra of RCE
photodetectors are peaked at the Fabry-Perot cavity resonances.
As in other
narrowband optical devices, the peak wavelengths are temperature
dependent.
Frequency demultiplexing photoreceivers for OFDM systems have
been proposed
using Bragg-reflector-based, frequency-selective elements [7].
However, the
frequency channels are split n times and then fed to n
photodetectors. Since all
channels are sent to each photodetector, the overall efficiency
of such a system is 1 n.
Furthermore, the fact that the optical power outside the
absorption band is reflected
presents a major problem. A large fraction, n −1( ) n, of the
optical power is reflectedback into the transmitter, adding demands
on isolation. A satisfactory scheme for
efficiently collecting light reflected from an RCE photodetector
and directing it to
other channels appears unlikely.
The waveguide photodetector (WGPD), schematically drawn in Fig.
1.4(a), is
another option for increasing the bandwidth-efficiency product
over the intrinsic limit
of conventional VPDs [1, 8]. The WGPD is an in-plane illuminated
photodetector in
which transparent dielectric cladding layers about the absorbing
core form a dielectric
optical waveguide [9-13]. The illumination is guided
perpendicular to the carrier drift
field, allowing a long absorption path while maintaining a small
junction area, so the
interdependence of bandwidth and internal efficiency is reduced.
High external
efficiency then depends on coupling most of the light into the
waveguide, which can
be accomplished by appropriate design of the device layers [14]
or by the use of
separate input waveguide segments [15-18].
-
6 1. Introduction
p
i
n
Absorber R E Light
(a)
R Z0
-
1.1 Background 7
R Z0 ≡ RI
Fig. 1.5 Simplified circuit diagram of a travelling-wave
photodetector. The travelling-wavephotodetector is a matched
electrical transmission line.
waveguide or connection, or there is a large impedance mismatch
between the
photodetector and the load, as suggested in the simplified
circuit diagram of Fig.
1.4(b). Multiple electrical reflections in the device cause the
entire junction area to
participate in the response. This is why WGPDs are best
represented by a lumped-
element model, as in Fig. 1.1, and suffer from an RC bandwidth
limitation in which
the device capacitance is determined by the total junction
area.
The travelling-wave photodetector (TWPD) is a fully distributed
structure that
overcomes the RC bandwidth limitation of the lumped-element
WGPD, while
retaining the advantages of the WGPD for optical bandwidth,
temperature range, and
integrability. It is based on the WGPD, but has an electrode
arrangement designed to
support travelling electrical waves with characteristic
impedance matched to that of the
external circuit, as indicated in the simplified circuit diagram
of Fig. 1.5 [28].
The TWPD is modelled by a terminated section of transmission
line with an
exponentially decaying photocurrent source propagating on it at
the optical group
velocity. The TWPD velocity-mismatch bandwidth limitation
depends on the optical
absorption coefficient and the mismatch between optical group
velocity and electrical
phase velocity, as opposed to an RC bandwidth limitation
determined by the total
junction area. TWPDs are not subject to the same RC bandwidth
limitation as
lumped-element photodetectors and can simultaneously have a
large bandwidth and
high efficiency.
The concept of the TWPD was first presented in 1990 as a means
to overcome the
bandwidth-efficiency limits of conventional photodetectors [29].
Both p-n and
Schottky junctions were briefly mentioned, although the several
designs listed in the
proposal were of the metal-semiconductor type. The bandwidths of
the proposed
structures, as depicted by the schematic drawings, would have
been severely limited
-
8 1. Introduction
by transit-time. No detailed theory or experimental results from
these designs have
been reported.
The use of doped layers generally results in a slow-wave
structure. However, in
1991, a proposal for a velocity-matched p-i-n TWPD was announced
[30]. A theory
of the effects of velocity matching on device response was not
included in this report.
This group is focussed on high-power, high-bandwidth
applications. Their efforts to
date have yielded devices with only 4.8 GHz bandwidth [31].
Unaware of other efforts, we began to investigate ways to
improve conventional
WGPDs at UCSB in 1991 and immediately conceived of the TWPD. We
were the
first to publish the basic theory of TWPDs, quantifying the
velocity mismatch
impulse response and associated bandwidth limitations [28]. We
subsequently
published the first experimental demonstration of TWPDs in 1994
[32]. The TWPDs
demonstrated significantly higher bandwidths and
bandwidth-efficiency products than
comparable WGPDs and VPDs on the same wafer [33]. After
correcting for the
measurement system response, these devices showed bandwidths as
high as
190 GHz and bandwidth-efficiency products as large as 84 GHz,
breaking the
records for bandwidth of a p-i-n photodetector by 70% and
bandwidth-efficiency
product of any photodetector without gain by 50%.
1 . 2 Range and Organization of this Dissertation
This dissertation covers the invention of the travelling-wave
photodetector,
including the first demonstration of a prototype device.
Analysis of the results show
reasonable correlation with the definition and description. The
conception, theory,
design, fabrication, measurement, and analysis of TWPDs are all
included in this
dissertation.
This introduction provides motivation and background for the
work, and it lays
out the structure of the subsequent presentation. High-speed
photodetector theory as
it relates to primary performance parameters is outlined. A
cursory description of
various types of photodetectors and their performance
characteristics is then followed
by the origins and brief history of the TWPD.
Chapter Two presents the theory that distinguishes the TWPD,
laying the
foundation for the rest of the dissertation. The model for a
fully distributed
-
1.2 Range and Organization of this Dissertation 9
photodetector is presented. The photodetector structure
electrical waveguide
properties are analyzed and the velocity-mismatch impulse
response is derived. The
velocity-mismatch effective area is deduced from the
velocity-mismatch frequency
response. This area facilitates the use of conventional methods
of photodetector
design and analysis on TWPDs. A simple theory of field-screening
effects, which
limit the high-illumination response of photodetectors, is
presented using empirical
parameters.
Chapter Three applies the theory presented in Chapter Two to the
design and
fabrication of TWPDs. A bandwidth model for a practical device
structure is
developed and used along with optoelectronic device physics,
processing, and
measurement considerations to optimize a device design. The
modified ridge-
waveguide laser process used to fabricate TWPDs is described
with special attention
to unique and newly developed features.
Chapter Four covers the construction of an electro-optic
sampling system, which
was adjunct to this thesis work. The system hardware and
principles are described
with special attention to new developments important for
high-speed photodetector
measurements.
Chapter Five presents measurement results with emphasis on
evidence of
TWPDs. Tavelling wave properties are elucidated and contrasted
to lumped element
characteristics of the WGPDs and VPDs. Post-measurement signal
processing
increases the range of measurement parameters over which
accurate analysis is
possible. Measurements of field-screening effects support the
theory presented in
Section 2.5.
Chapter Six summarizes this work and draws some conclusions.
Successes and
shortcomings are discussed, providing seeds for future
directions. Natural extensions
of the TWPD concept and applications of the technology are
proposed.
Appendix A lists a Matlab program used for calculating results
presented in
Sections 2.1 and 2.2 [34]. Appendix B details the TWPD
fabrication process
described in Section 3.4.
-
10 1. Introduction
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Long-Wavelength p-i-n Photodetectors,"J. Lightwave Technol., vol.
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[3] A. Chin and T. Y. Chang, "Enhancement of Quantum Efficiency
in Thin PhotodiodesThrough Absorptive Resonance," J. Lightwave
Technol., vol. 9, no. 3, pp. 321-328, Mar.,1991.
[4] K. Kishino, M. S. Ünlü, J. Chyi, J. Reed, L. Arsenault, and
H. Morkoç, "Resonant Cavity-Enhanced (RCE) Photodetectors," IEEE J.
Quantum Electron., vol. 27, no. 8, pp. 2025-2034,Aug., 1991.
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[10] J. L. Merz and R. A. Logan, "Integrated GaAs-AlxGa1-xAs
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[12] J. E. Bowers and C. A. Burrus, "High-Speed Zero-Bias
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905-906, Aug. 14, 1986.
[13] D. Wake, T. P. Spooner, S. D. Perrin, and I. D. Henning,
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27, no. 12, pp. 1073-1075, Jun. 6, 1991.
[14] K. Kato, S. Hata, K. Kawano, J. Yoshida, and A. Kozen, "A
High-Efficiency 50 GHzInGaAs Multimode Waveguide Photodetector,"
IEEE J. Quantum Electron., vol. 28, no. 12,pp. 2728-2735, Dec.,
1992.
[15] Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and
D. A. Ackerman, "Efficientcoupling of a semiconductor laser to an
optical fiber by means of a tapered waveguide onsilicon," Appl.
Phys. Lett., vol. 55, no. 23, pp. 2389-2391, Dec. 4, 1989.
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References 11
[16] T. Brenner and H. Melchior, "Integrated Optical Modeshape
Adapters in InGaAsP/InP forEfficient Fiber-to-Waveguide Coupling,"
IEEE Photon. Technol. Lett., vol. 5, no. 9, 1053-1056, Sep.,
1993.
[17] S. El Yumin, K. Komori, and S. Arai, "GaInAsP/InP
Semiconductor Vertical GRIN-Lens forSemiconductor Optical Devices,"
IEEE Photon. Technol. Lett., vol. 6, no. 5, pp. 601-604,May,
1994.
[18] I. Moerman, M. D'Hondt, W. Vanderbauwhede, G. Coudenys, J.
Haes, P. De Dobbelaere, R.Baets, P. Van Daele, and P. Demeester,
"Monolithic Integration of a Spot Size Transfromerwith a Planar
Buried Heterostructure in GaAsP/InP-Laser using the Shadow Masked
GrowthTechnique," IEEE Photon. Technol. Lett., vol. 6, no. 8, pp.
888-890, Aug., 1994.
[19] K. Kato, A. Kozen, Y. Muramoto, Y. Itaya, T. Nagatsuma, and
M. Yaita, "110-GHz, 50%-Efficiency Mushroom-Mesa Waveguide p-i-n
Photodiode for a 1.55-µm Wavelength," IEEEPhoton. Technol. Lett.,
vol. 6, no. 6, 719-721, Jun., 1994.
[20] R. J. Deri, "Monolithic Integration of Optical Waveguide
Circuitry with III-V Photodetectorsfor Advanced Lightwave
Receivers," J. Lightwave Technol., vol. 11, no. 8, pp.
1296-1313,Aug., 1993.
[21] D. Wake, "A 1550-nm Millimeter-Wave Photodetector with a
Bandwidth-Efficiency Product of2.4 THz," J. Lightwave Technol.,
vol. 10, no. 7, pp. 908-912, Jul., 1992.
[22] P. A. Kirby, "Multichannel Wavelength-Switched Transmitters
and Receivers– NewComponent Concepts for Broad-Band Networds and
Distributed Switching Systems," J.Lightwave Technol., vol. 8, no.
2, pp. 202-211, Feb., 1990.
[23] W. J. Grande, J. E. Johnson, C. L. Tang, "GaAs/AlGaAs
photonic integrated circuitsfabricated using chemically assisted
ion beam etching," Appl. Phys. Lett., vol. 57, no. 24,pp.
2537-2539, Dec. 10, 1990.
[24] C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus,
and L. Stoll, "GratingSpectrograph Integrated with Photodiode Array
in InGaAsP/InGaAs/InP," IEEE Photon.Technol. Lett., vol. 4, no. 1,
pp. 108-110, Jan., 1992.
[25] P. C. Clemens, G. Heise, R. März, H. Michel, A. Reichelt,
and H. W. Schneider, "8-ChannelOptical Demultiplexer Realized as
SiO2/Si Flat-Field Spectrograph," IEEE Photon. Technol.Lett., vol.
6, no. 8, pp. 1109-1111, Sep., 1994.
[26] C. Dragone, "An NxN Optical Multiplexer Using a Planar
Arrangement of Two StarCouplers" IEEE Photon. Technol. Lett., vol.
3, no. 9, pp 812-815, Sep., 1991.
[27] C. Dragone, C. A. Edwards, and R. C. Kistler, "Integrated
Optics NxN multiplexor inSilicon" IEEE Photon. Technol. Lett., vol.
3, no. 10, pp 896-899, Oct., 1991.
[28] K. S. Giboney, M. J. W. Rodwell, and J. E. Bowers,
"Traveling-Wave Photodetectors," IEEEPhoton. Technol. Lett., vol.
4, no. 12, pp. 1363-1365, Dec., 1992.
[29] H. F. Taylor, O. Eknoyan, C. S. Park, K. N. Choi, and K.
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-
12 1. Introduction
[30] V. M. Heitala and G. A. Vawter, "A Large-Bandwidth
High-Quantum-Efficiency Traveling-Wave Photodetector Based on a
Slow-Wave Coplanar Transmission Line," presented at
Prog.Electromagnetics Res. Symp., Cambridge, MA, Jul., 1991.
[31] V. M. Hietala, G. A. Vawter, T. M. Brennan, and B. E.
Hammons, "Traveling-WavePhotodetectors for High-Power Large
Bandwidth Applications," IEEE Trans. MicrowaveTheory Tech.,
submitted, 1995
[32] K. Giboney, R. Nagarajan, T. Reynolds, S. Allen, R. Mirin,
M. Rodwell, and J. Bowers,"172 GHz, 42% Quantum Efficiency p-i-n
Travelling-Wave Photodetector," presented at 52ndAnnual Device Res.
Conf., Boulder, CO, Jun., 1994, VIA-9.
[33] K. S. Giboney, R. L. Nagarajan, T. E. Reynolds, S. T.
Allen, R. P. Mirin, M. J. W.Rodwell, and J. E. Bowers,
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Bandwidth-Efficiency Product," IEEE Photon. Technol. Lett., vol. 7,
no. 4, pp. 412-414, Apr., 1995.
[34] The MathWorks, Inc., The Student Edition of Matlab Version
4. Englewood Cliffs, NJ:Prentice Hall, 1995.
-
13
CHAPTER 2
THEORY OF DISTRIBUTED PHOTODETECTION
A travelling-wave photodetector (TWPD) is generally much shorter
than the
electrical wavelengths that it is designed for, but much longer
than its optical
absorption length. The interaction of the optical and electrical
waves, or
photodetection, occurs over the relatively short optical
absorption length. This length
and the electrical and optical wave velocities determine the
device response. Thus a
distributed, rather than lumped, model applies.
The TWPD is based on the waveguide photodetector (WGPD). The
characteristics of the TWPD that distinguish it from the WGPD
are a TEM or quasi–
TEM electrical waveguide that is concomitant with the optical
dielectric waveguide,
and a matched electrical termination at the device output end.
These features provide
for controlled transmission of the electrical wave down the
device in parallel with the
optical wave and eliminate bandwidth-limiting reflections.
The potential for copropagation of electrical and optical waves
in a photodetector
naturally presents the prospect of designing travelling-wave
device. A travelling-wave
structure is defined when two or more distinct waves, optical,
electrical, acoustical,
or other, interact over some distance as they propagate down
their respective
waveguides. If the interaction is coherent over the distance of
interest, then the device
can be said to be velocity-matched. The distance of interest in
a TWPD is determined
by the optical absorption.
TWPDs are generally not velocity-matched. Electrical waves
propagate in the
slow-wave mode and are usually drastically slower than the
optical waves generating
them. This results in a distortion of the signal that is the
origin of the velocity-
mismatch bandwidth limitation. The velocity-mismatch impulse
response is derived
directly from the optical group and electrical phase velocities
and the optical
absorption. A simple form for the velocity-mismatch bandwidth
limitation allows
determination of the overall photodetector bandwidth by
traditional methods. It is
-
14 2. Theory of Distributed Photodetection
pin
z
x
y
w
d
D
Light
llll
Ri
Ro ≡≡≡≡ Z0
Fig. 2.1 Fully distributed, parallel-plate, p-i-n TWPD schematic
diagram. The termination at theoutput end is defined to match the
characteristic impedance of the device.
clear from this form that the velocity-mismatch bandwidth
limitation is nearly
independent of TWPD length.
Photodetector performance under high-power illumination is
important for many
applications. High photogenerated charge densities cause
nonlinear response in high-
speed photodetectors primarily by field screening, resulting in
distorted output and
reduced bandwidth. A heuristic theory explains how TWPDs take
advantage of a
mechanism by which in-plane illuminated photodetectors adapt to
high-power
illumination.
2 . 1 Photodetector-Electrical Waveguide
The TWPD is an electrical and optical waveguide. The fully
distributed TWPD
structure drawn in Fig. 2.1 allows an analytical representation,
and offers insight into
the electrical propagation characteristics. In addition to
forming the photodiode, the
metal-clad, p-i-n structure is a parallel-plate electrical
waveguide, and the double-
heterostructure semiconductor layers form a planar dielectric
optical waveguide. In
this section, the electrical propagation characteristics of this
structure are analyzed
-
2.1 Photodetector-Electrical Waveguide 15
Semiconductor
DepletionInsulator
Semiconductor
x
y
Metal
Metal
Metal
Metal
x
(a) (b)
Fig. 2.2 Metal-insulator-semiconductor (MIS) structures for (a)
integrated circuit interconnects and(b) Schottky contact
transmission line.
from the wave equation using the transverse-resonance method.
Losses associated
with finite metal conductivity are accounted for.
The TWPD structure is similar to metal-insulator-semiconductor
(MIS) structures
that have been extensively analyzed [1-17]. MIS structures, such
as those drawn in
Fig. 2.2, are of interest because they arise when interconnects
or Schottky contacts
are fabricated on semiconducting substrates. The properties of
transmission lines
formed in this manner are of consequence for a wide range of
integrated circuits and
devices. They are called "slow-wave" transmission lines because
the phase velocities
of supported modes are much slower than expected simply from the
permittivities and
permeabilities of the media.
The parallel-plate configuration of Fig. 2.1 allows accurate
two-dimensional
analysis [1-3]. More complicated structures nominally require
three-dimensional
analyses [4-12], however certain symmetries often permit an
effective dimensional
reduction [13-17]. Such approaches are discussed in Chapter 3 in
conjunction with
TWPD design and fabrication.
The structure of the TWPD differs from that of an MIS structure
by an additional
semiconductor layer, however the general characteristics of the
waves supported on
these structures are similar. This is seen by assuming that the
p- and n-layers of the
-
16 2. Theory of Distributed Photodetection
Semiconductor
Insulator
x
z
Metal
Electric Wall
Air
di 2
ds
dm
ηxi
ηxs
ηxm
ηx0
ηxm0ηx+ ηx−dc ηxcContact
(a) (b)
Fig. 2.3 TWPD half-waveguide (a) layer structure and (b)
transmission line equivalent circuit fortransverse waves.
TWPD are of equal conductivity and thickness, and then applying
image theory. The
waveguide is cut in half, as shown in Fig. 2.3(a), with an
electric wall at the plane of
symmetry. The TWPD now looks like an MIS structure with a
perfect conductor as
the top metal. A thin (10 nm) contact layer is included in the
model of Fig. 2.3 to
simulate the metal-semiconductor junction.
This structure is analyzed by starting with the time-harmonic
form of Maxwell's
equations in linear media, using cosine-based phasors
∇ × E = − jωµH (2.1a)
∇ × H = jωεE . (2.1b)
Low-loss dielectrics are assumed and conductivity is
incorporated into the dielectric
constant: ε = ε '− j σ ω . The wave equation follows directly
from Eqs. (2.1)
∇2H + ω 2µεH = 0. (2.2)
-
2.1 Photodetector-Electrical Waveguide 17
Electromagnetic waves will propagate only in the
transverse-magnetic (TM) mode up
to frequencies far above 1 THz in micron-dimension waveguides.
Assuming no
variation in the y-direction, the H-field is purely y-directed.
The H-field for the nth
layer is then,
Hn = ŷ H++ne
− j kxnx+kzz( ) + H+−ne− j kxnx−kzz( ) + H−+ne
j kxnx−kzz( ) + H−−nej kxnx+kzz( )[ ] (2.3)
where H represents the wave component amplitude, kxn is the
transverse propagation
constant in the nth layer, and kz is the longitudinal
propagation constant.
kn
2 ≡ ω 2µnεn = kxn2 + kz
2 is the dispersion relation. The transverse wave impedance
of
the nth layer is
ηxn = kxn ωεn . The electric fields are found directly from
(2.1b).
A set of waves in the form of (2.3) satisfying the boundary
conditions of the
structure will propagate longitudinally (in the z-direction) and
resonate transversely
(in the x-direction). The transverse resonance condition is used
to find the
longitudinal propagation constant, kz, which characterizes the
mode. Methods from
mathematically equivalent, and perhaps more familiar,
transmission line theory apply,
and the problem can be analyzed using the transmission line
circuit analog shown in
Fig. 2.3(b).
The net impedance of the metal-air layers is reduced to a simple
expression by
assuming the wave impedance in the metal is much less than the
transverse waveimpedance in air,
ηm = jωµ0 σm
-
18 2. Theory of Distributed Photodetection
10-3
10-2
10-1
100
101
0 0.2 0.4 0.6 0.8 1 1.2 1.4
H-F
ield
Mag
nitu
de (
AU
)
Depth, x (µm)
10 GHz
100 GHz
ρs = 100,1,2,3 Ω ·µm
Semiconductor Metal
Insulator Contact
Air
Fig. 2.4 Magnetic field distribution in a parallel-plate TWPD at
10 GHz and 100 GHz for a rangeof semiconductor layer resistivities
of 1 to 1,000 Ω ·µm. The traces are coincident for all of
theplotted semiconductor layer resistivities.
ηxcηxm0 + jηxc tan kxcdc( )ηxc + jηxm0 tan kxcdc( )
+ jηxsηxi tan kxidi 2( ) + ηxs tan kxsds( )ηxs − ηxi tan kxidi
2( ) tan kxsds( )
= 0. (2.5)
This equation is solved numerically for kz via the dispersion
relation. A Matlab
program that solves (2.5) and computes the fields and
propagation characteristics is
listed in Appendix A [18]. The electrical phase velocity and
field attenuation constant
are found from the real and imaginary parts of kz. A waveguide
width is assumed for
the characteristic impedance, which is directly proportional to
kz. The waveguide
dimensions are chosen such that a one micron wide waveguide will
have a
characteristic impedance of about 50 ohms.
Field distributions in the waveguide are plotted in Figs. 2.4 –
2.6. The H-field
magnitude at 10 GHz and 100 GHz is plotted in Fig. 2.4. This
plot shows that most
of the current is carried in the metal layers, although the
penetration into the metal
decreases with increasing frequency. The current in the
semiconductor layers
-
2.1 Photodetector-Electrical Waveguide 19
10-6
10-4
10-2
100
102
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E-F
ield
Mag
nitu
de (
AU
)
Depth, x (µm)
ρs = 100 Ω ·µm
Ez
Ex
10 GHz100 GHz
Semiconductor Metal
Insulator Contact
Air
Ez
Ex
Fig. 2.5 Electric field components in a parallel-plate TWPD at
10 GHz and 100 GHz forsemiconductor layer resistivity of 100
Ω·µm.
increases slightly with frequency. Varying the resistivity of
the semiconductor layers
over three orders of magnitude produces little effect on the
overall current distribution
in the device.
The magnitudes of both E-field components at 10 GHz and 100 GHz
are plotted
in Fig. 2.5 for a semiconductor resistivity of 100 Ω·µm, which
is a typical resistivityfor heavily doped semiconductor. Most of
the voltage is supported by the insulator
layer, although the transverse fields in the semiconductor and
metal layers increase
with increasing frequency. The longitudinal fields in the
insulator and semiconductor
layers also increase with increasing frequency. The change in
longitudinal fields in
the metal layers reflects the changing current distribution
inferred from Fig. 2.4.
Fields in the semiconductor and metal layers produce
longitudinal and transverse
currents, causing attenuation on the waveguide.
Fig. 2.6 shows the E-field components at 100 GHz for various
semiconductor
resistivities. The x-component in the semiconductor approaches
the value in the metal
as the resistivity is decreased, as expected. The z-component
doesn't change
-
20 2. Theory of Distributed Photodetection
10-6
10-4
10-2
100
102
0 0.2 0.4 0.6 0.8 1 1.2 1.4
E-F
ield
Mag
nitu
de (
AU
)
Depth, x (µm)
100
101
102
103
ρs (Ω ·µm) f = 100 GHz
Ez
Ex
Semiconductor Metal
Insulator Contact
Air
Ez
Ex
Fig. 2.6 Electric field components in a parallel-plate TWPD at
100 GHz for semiconductor layerresistivities of 1 to 1,000
Ω·µm.
significantly with semiconductor resistivity. Note that the x-
and z-components are
nearly equal in the semiconductor layers when the semiconductor
resistivity is about
10 Ω·µm. This is approximately a point of minimum attenuation,
evident in the plotof field attenuation constant versus
semiconductor resistivity in Fig. 2.7. Also shown
for comparison are plots of the attenuation constant of a
waveguide with perfect metal
rather than gold, and of a gold MIM (metal-insulator-metal)
waveguide.
2 . 2 Equivalent-Circuit ModelThe fact that
Ex is much larger than Ez in the insulator, across which most of
the
voltage appears, together with the fact that the mode is TM
establish that propagation
on the TWPD is approximately transverse-electromagnetic, or
quasi-TEM. A
transmission line equivalent-circuit model accurately describes
the properties of a
quasi-TEM waveguide.
Such a circuit for a TWPD is shown in Fig. 2.8, and the element
values for a
parallel-plate TWPD are listed in Table 2.1. Fig. 2.8(a) shows
the physical origins of
-
2.2 Equivalent-Circuit Model 21
10-4
10-3
10-2
100 101 102 103 104
Fie
ld A
ttenu
atio
n C
onst
ant
(µm
-1)
Resistivity (Ω·µm)
f = 100 GHz
0.48 (MIM)
0.2
di (µm)
Gold MetalPerfect Metal
Fig. 2.7 Field attenuation constant as a function of
semiconductor layer resistivity at 100 GHz.Curves for gold and
perfect conductor metal layers with insulator layer thicknesses of
0.2 µm and agold metal-insulator-metal (MIM) waveguide with
insulator layer thickness of 0.48 µm (givingcharacteristic
impedance of 50 Ω for a 1 µm wide waveguide) are shown.
the circuit elements. Conduction and displacement currents in
the contact and
semiconductor layers are accounted for by resistances in
parallel with capacitances, as
shown in Fig. 2.8(b). At frequencies far below the dielectric
relaxation frequencies ofthe contact and semiconductor layers (
ωρcεc
-
22 2. Theory of Distributed Photodetection
Semiconductor
Insulator
x
z
Metal
Electric Wall
Air
Contact
Zm
Ci
Lm
Gs
Zc
Zs
(a)
Zc,Zs Rc,Rs Cc,Cs
(b)
Fig. 2.8 TWPD equivalent-circuit models for (a) transmission
line characteristics and (b)semiconductor and contact layer
impedances. Circuit elements are identified with their
associatedlayers from Fig. 2.3. Element values for a parallel-plate
TWPD are listed in Table 2.1.
conductance of the semiconductor layers, Gs, includes a divisor
of three to account
for the current distribution in the semiconductor layers
[3].
The transmission line propagation characteristics may be found
by comparing the
equivalent-circuit model of Fig. 2.8 to the general transmission
line equivalent-circuit
model of Fig. 2.9. Coupled differential equations for voltage
and current wave
propagation are written in terms of series impedance, Z, and
shunt admittance, Y, per
unit length of transmission line
-
2.2 Equivalent-Circuit Model 23
Zc =ρc
1+ jωρcεc⋅ 2dc
w Zm = ηxm0
2w
Zs =ρs
1+ jωρsεs⋅ 2ds
w Gs =
σs3
⋅ wds2
Ci = ε iw
di Lm = µ0
D
w
Table 2.1 Parallel-plate TWPD element values for the
equivalent-circuit model in Fig. 2.8. ρ isresistivity and σ = 1 ρ
is conductivity; ε and µ are permittivity and permeability. The
metal-airtransverse wave impedance,
η
xm0, is defined Fig. 2.3(b) and expressed in Eq. (2.4). The
dimensions
are indicated in Figs. 2.1 and 2.3(a).
Z
Y
Fig. 2.9 General transmission line equivalent-circuit model.
dV
dz= − IZ (2.6a)
dI
dz= −VY. (2.6b)
These equations lead directly to the wave equation,
d2V
dz2= γ 2V (2.7)
where
γ = YZ = αe + jβ (2.8)
is the complex propagation constant of the general solution, V
z( ) = V0+e−γz + V0−eγz .
Microwave loss and propagation velocity are found directly from
the field attenuation
-
24 2. Theory of Distributed Photodetection
0
20
40
60
80
100
0.1 1 10 100 1000
Ele
ctric
al P
hase
Vel
ocity
(µm
/ps)
Frequency (GHz)
10100
1000
ρs (Ω ·µm)
di = 0.2 µm
Transverse ResonanceEquivalent Circuit
Fig. 2.10 Parallel-plate TWPD electrical velocity vs. frequency
from transverse resonance solutionand equivalent circuit model for
semiconductor layer resistivities of 10, 100, and 1000 Ω·µm.
Thefull thickness of the insulator layer is 0.2 µm.
constant, αe, and the propagation or phase constant, β. The
voltage and current
waves propagating on the transmission line are related by the
characteristic
impedance,
Z0 = Z Y. (2.9)
The electrical wave velocity, field attenuation constant, and
characteristic
impedance calculated from the equivalent-circuit model are
compared with those from
the transverse resonance solution in Figs. 2.10 – 2.12 for
semiconductor resistivities
of 10, 100, and 1000 Ω·µm. Complex impedances for the contact
and semiconductorlayers in the equivalent-circuit model enable
excellent accuracy for the propagation
constant to beyond 1 THz. The equivalent-circuit model predicts
characteristic
impedance very well, also, aside from some deviation at high
frequencies in the
ρs = 1000 Ω·µm traces. The equivalent-circuit model accuracy
shown in these plotsholds with variations in the insulator layer
thickness, as well. Thus, the equivalent-
circuit model can be considered accurate to 1 THz for the
structures of interest here.
-
2.2 Equivalent-Circuit Model 25
10-4
10-3
10-2
10-1
0.1 1 10 100 1000
Fie
ld A
ttenu
atio
n C
onst
ant
(µm
-1)
Frequency (GHz)
di = 0.2 µm
Transverse ResonanceEquivalent Circuit
10100
1000
ρs (Ω ·µm)
Fig. 2.11 Parallel-plate TWPD field attenuation constant,
comparing transverse resonance solutionand equivalent circuit
model.
Further analysis from the equivalent-circuit model yields some
insight into the
behavior seen in Figs. 2.10 – 2.12. Assuming the metal
conductivity is largecompared to the parallel conductance of the
semiconductor layers,
GZm
-
26 2. Theory of Distributed Photodetection
-200
-100
0
100
200
0.1 1 10 100 1000
Cha
ract
eris
tic I
mpe
danc
e ( Ω
)
Frequency (GHz)
100010010
ρs (Ω ·µm)
w = 1 µmd
i = 0.2 µm
Re
Im
Transverse ResonanceEquivalent Circuit
Fig. 2.12 Characteristic impedance of a 1 µm wide,
parallel-plate TWPD from transverse resonancesolution and
equivalent circuit model.
G
L
R
C
G
L
R
C I1 I2
Zm Zm
RoRi
Fig. 2.13 TWPD equivalent-circuit diagram.
The corresponding frequency range in the plots is below about 1
GHz. The highly
dispersive propagation in this range is characteristic of
diffusion. This is typical of
transmission lines at very low frequencies due to the dominance
of the metal
impedance.
-
2.2 Equivalent-Circuit Model 27
The propagation characteristics in the mid-frequency range ( ωRC
1) are of greatest interest for TWPDs, since the devicebandwidth
falls in this range. The complex propagation constant can be
written
γ ≈ ω2
2veRC+ GL( ) + j ω
ve(2.12)
where ve ≡ ω β = 1 LC . The loss is proportional to the square
of frequency, and
consists of terms proportional to RC and GL. The minimum in Fig.
2.7 occurs when
RC = GL. The characteristic impedance is
Z0 ≈ R0 1+ jω2
RC− GL( )
(2.13)
where R0 ≡ L C . The effect of RC and GL on the characteristic
impedance is to add
a small reactance, which disappears when RC = GL. Notice that
the metal impedancedoes not significantly affect the propagation
characteristics when
ωL Zm >> 1.
The mid-frequency range covers from about 10 GHz to several
hundred gigahertz
in Figs. 2.10 – 2.12. It is clear from (2.12) and (2.13) that
the standard lossless
transmission line equations for phase velocity and
characteristic impedance,
ve = 1 LC (2.14)
Z0 = L C (2.15)
are good approximations for TWPDs over this frequency range.
At high frequencies, the more detailed transmission line
equivalent-circuit model
of Fig. 2.8 applies, given the semiconductor layer is much
thinner than its skin depth.
This is true up to 1 THz for the TWPD designs considered here.
Far above thedielectric relaxation frequencies of the contact and
semiconductor layers (
ωρcεc >> 1
and ωρsεs >> 1), the overall transmission line capacitance
is the capacitance of all of
the layers between the metal layers, Cm = 1 Ci +1 Cs +1 Cc( )−1,
and the transverse
resistance of those layers, R= Rc + Rs, is negligible. At high
frequencies
( ωρcεc >> 1, ωρsεs >> 1, and ωGsLm
-
28 2. Theory of Distributed Photodetection
γ ≈ ω2
2vemGsLm + j
ωvem
(2.16)
Z0 ≈ R0m 1− jω2
GsLm
(2.17)
where vem ≡ 1 LmCm and R0m ≡ Lm Cm .
In the high frequency regime, the voltage and current are not
spatially separated,
so the phase velocity and characteristic impedance are not low,
as they are at lower
frequencies. Additionally, the attenuation is lower due to the
elimination of the RCterm in the attenuation constant. These
effects explain the different trends of the
ρs =
1000 Ω ·µm traces in Figs. 2.10 – 2.12 above a few hundred
gigahertz.Characteristics of this mode are similar to those of TEM
propagation in a lossy
dielectric, so it is sometimes called the dielectric quasi-TEM
mode [1, 2].
2 . 3 Velocity-Mismatch Impulse Response
The TWPD is a length of matched transmission line with a
position-dependent
photocurrent source distributed along its length, as represented
by the equivalent-
circuit diagram in Fig. 2.13. The TWPD velocity-mismatch impulse
response is
determined by the mismatch between optical group velocity and
electrical phase
velocity, and by the optical absorption coefficient [19]. In
contrast, the RC bandwidth
limitation of a lumped-element device, such as a WGPD, is
determined by the total
junction area [20]. The TWPD characteristic impedance is matched
to the load
impedance, so electrical waves are not reflected at the load as
they are in lumped-
element devices. This gives the TWPD a distinct impulse response
that is essentially
independent of device length.
The velocity mismatch is generally very large in a fully
distributed TWPD because
electrical waves propagate in the slow-wave mode over the
frequency range of
interest, as explained in Sections 2.1 and 2.2. The optical
group and electrical phase
velocities in a TWPD are mismatched by a factor of about 3:1
over the i-layer
thickness range for 100-200 GHz bandwidth operation, as
highlighted in the plot of
Fig. 2.14. Light is absorbed in the i-layer, generating
electrical waves, as the optical
wave propagates on the structure. The impulse response resulting
from these effects,
-
2.3 Velocity-Mismatch Impulse Response 29
0
20
40
60
80
100
0 50 100 150 200 250
Vel
ocity
(µm
/ps)
vo
ve
3:1
i-Layer Thickness (nm)
Fig. 2.14 Electrical phase velocity and optical group velocity
calculated for a GaAs/AlGaAsparallel-plate TWPD.
and no others, is derived here. Accordingly, the carrier transit
times are assumed to
be zero.
Consider an impulsive packet of photons travelling in the
semiconductor
waveguide of a TWPD. The density of photons is given by
nph z,t( ) = Nph z( )δ z− vot( ) (2.18)
where Nph is the photon number, δ is the Dirac delta function,
and vo is the optical
group velocity. The probability of loss of a photon by linear
mechanisms such as
absorption and scattering is a constant, independent of
position, so the change in
photon number with distance is proportional to the photon
number,
dNph z( ) dz= −ΓαNph z( ) . The photon number then decays
exponentially in thedevice according to
Nph z( ) = Nph 0( )e−Γαz, where Γ is the optical waveguide
confinement factor, and α is the i-layer material optical power
absorption coefficient,which includes all linear loss mechanisms
(presumably dominated by absorption with
the generation of electron-hole pairs).
-
30 2. Theory of Distributed Photodetection
There is a direct correspondence between the absorption of
photons and the
generation of e-h pairs. The density of photogenerated e-h pairs
is proportional to thedistance rate of change of the photon
number,
neh z( ) = −ηi dNph z( ) dzu z( ), where ηi
is the fraction of lost photons generating e-h pairs that are
collected, and u is the unit
step function. The density of photogenerated e-h pairs is then
simply
neh z( ) = Γαηi Nph 0( )e−Γαzu z( ).
Electrical waves propagate in both directions on the waveguide,
originating from
the travelling photocurrent source. Presently assume that the
electrical phase velocityis very much smaller than the optical
group velocity,
ve
-
2.3 Velocity-Mismatch Impulse Response 31
To consider the case when the electrical velocity is not much
smaller than the
optical velocity, it is helpful to start in an inertial frame of
reference of an electrical
wave. The equation for the total forward-going charge density
wave is of the same
form as (2.19) when viewed in the forward-going electrical wave
frame of reference,
ρ z',t( ) = Q
2Γα 'e−Γα 'z'u z'( ) + γΓα ' 'eΓα ''z'u −z'( )[ ] (2.22)
where z'= z− vet , α '= α 1− ve vo( ) , and α ' '= α 1+ ve vo( )
. Converting back to the
device rest frame gives the forward-travelling charge density
wave
ρ z,t( ) = Q2
Γα1− ve vo
e− Γα
1−ve voz−vet( )
u z− vet( ) + γ Γα1+ ve voe
Γα1+ve vo
z−vet( )u vet − z( )
(2.23)
Inserting (2.23) into (2.20) and accounting for the (unusual)
case when ve > vo,
gives the desired expression for the TWPD velocity-mismatch
impulse response,
ivm t +l
ve
= Q2
ω f eω f tu −ω f t( ) + γω re−ω r tu t( )[ ] (2.24)
for min l vo,l ve( ) ≤ t ≤ l vo + 2l ve and zero elsewhere,
and
ω f = Γαve 1− ve vo( ) (2.25a)
ω r = Γαve 1+ ve vo( ) (2.25b)
are characteristic frequencies of the forward and reverse waves.
Equation (2.24) is a
general description of the current response of a TWPD to an
optical impulse applied at
its input considering only velocity mismatch.
-
32 2. Theory of Distributed Photodetection
ve
ve
ve
vo
LightImpulse
0 l(Input) (Output)
t = l4vo
Ph
oto
curr
en
t
ve
ve
ve
vo
LightImpulse
0 l
t = l2vo
(Input) (Output)
Ph
oto
curr
en
t
Distance0 l
ve
ve
ve
LightImpulse(Small)
t = lvo
(Input) (Output)
Ph
oto
curr
en
t
Fig. 2.15 Spatial plots of the velocity-mismatch impulse
response when the light impulse is atl/4 (top plot), in the middle
(center plot), and at the output end (bottom plot) of a TWPD with
open-circuit input termination.
-
2.3 Velocity-Mismatch Impulse Response 33
0 1 2 3 4 5
ForwardWave
ReflectedWave
Pho
tocu
rren
t (A
U)
Time (ps)
Fig. 2.16 Velocity mismatch impulse response of a TWPD with
open-circuit input termination for
v
o = 86 µm/ps,
v
e = 29 µm/ps, Γα = 0.1 /µm, and l = 50 µm.
Fig. 2.15 depicts the propagation of a light impulse and the
photogenerated
electrical waves on a TWPD, showing the formation of the
velocity-mismatch
impulse response in space. The temporal velocity-mismatch
impulse response from
(2.24) is plotted in Fig. 2.16. The response is composed of two
exponential
components, which in general have different decay constants.
Conservation of energy
and charge require that the integral of the photocurrent
response of an infinitely long
TWPD with γ = 1 equal the total charge available in the optical
pulse,
Q = Nph 0( )ηiq = E0ηiq hν , where E0 is the total energy in the
optical impulse.Furthermore, conservation of momentum for the
electrical waves dictates that the total
charge be split equally between the forward- and
reverse-travelling photocurrent
components. As a result, the heights of these two components are
generally different
where they meet, forming a discontinuity in the photocurrent
response at that point.
The velocity-mismatch impulse response can be made much shorter
by placing a
matched termination at the input end of the TWPD ( γ = 0). The
reflected wave is thenabsorbed and only the first term in (2.24)
remains. Half of the photocurrent is lost in
-
34 2. Theory of Distributed Photodetection
the termination, however, casting doubt on the value of this
approach. This is best
evaluated in the frequency domain.
2 . 4 Velocity-Mismatch Bandwidth Limitation
The velocity-mismatch frequency response is the Fourier
transform of the impulse
response, (2.24). For long TWPDs, Γαl >> 1( ), this is
given by
ivm ω( ) =Q
2
ω fω f − jω
+ γ ω( ) ω rω r + jω
e
− jω lve . (2.26)
The fractional photocurrent magnitude frequency response for
real values of γ is
ivm ω( )Q
2
= 14
⋅ω f − γω r( )2ω 2 + 1+ γ( )2ω f2ω r2
ω 2 + ω f2( ) ω 2 + ω r2( ) . (2.27)
In a TWPD with matched input termination ( γ = 0), only the
forward-travelling wavecontributes to the response since the
reverse-travelling wave is absorbed in the input
termination. The magnitude photocurrent response (2.27) for γ =
0 reduces to thesingle-pole response,
ivm0 ω( )Q
2
= 14
⋅ 11+ ω ω f( )2
(2.28)
and the 3 dB bandwidth limitation is Bvm0 = ω f 2π .
-
2.4 Velocity-Mismatch Bandwidth Limitation 35
0
0.5
1
0 1 2 3 4
Fra
ctio
nal P
hoto
curr
ent,
|i(ω )
/Q|
Normalized Frequency, ω/Γ αve
γ = 1
γ = 0
ve vo = 0.33
Fig. 2.17 Velocity mismatch frequency response for TWPDs with
large velocity mismatch,
v
ev
o= 0.33, for open-circuit ( γ = 1) and matched ( γ = 0) input
terminations.
Fig. 2.17 shows velocity-mismatch frequency response curves for
a TWPD witha typical velocity mismatch,
ve vo = 0.33, for open-circuit ( γ = 1) and matched
( γ = 0) input terminations. The quantum efficiency of the
device with matched inputtermination is limited to 50%. In this
particular case, the γ = 1 curve intersects the
γ = 0 curve at the -3 dB point of the γ = 0 response, so the
TWPD response withopen-circuit input termination ( γ = 1) is larger
than that of the TWPD with matchedinput termination ( γ = 0) over
its entire bandwidth.
The TWPD with matched input termination ( γ = 0) has no
bandwidth limitationin the velocity-matched case,
ve vo = 1, but is still limited to 50% efficiency, as
shown in Fig. 2.18. The γ = 1 curve is always above the γ = 0
curve in this plot.The bandwidth of TWPDs are limited by other
factors, also (discussed in Chapter 3),
and the net bandwidth-efficiency product is generally worse in
devices with matched
input terminations [19].
-
36 2. Theory of Distributed Photodetection
0
0.5
1
0 1 2 3 4
Fra
ctio
nal P
hoto
curr
ent,
|i(ω )
/Q|
Normalized Frequency, ω/Γ αve
γ = 1
γ = 0
ve vo =1
Fig. 2.18 Velocity mismatch frequency response for velocity
matched ( v
ev
o= 1) TWPDs for
open-circuit ( γ = 1) and matched ( γ = 0) input
terminations.
Comparing the γ = 1 curves in Figs. 2.17 and 2.18, it appears
that velocitymatching has little effect on their 3 dB bandwidths.
The bandwidth for positive real
values of γ can be found by solving a quadratic equation for
ω1,
ω14 + ω f
2 + ω r2 − 2
1+ γ( )2ω f − γω r( )2
ω1
2 − ω f2ω r
2 = 0. (2.29)
Fig. 2.19 confirms that the γ = 1 curve is almost independent of
velocity mismatch,while the γ = 0 curve is strongly dependent.
The response of a γ = 1 TWPD is composed of forward- and
reverse-travellingwaves. While velocity matching reduces the
temporal duration of the forward-
travelling wave, it stretches the reverse-travelling wave, so
the overall duration of the
γ = 1 response changes little. The velocity-mismatch bandwidth
limitation for a γ = 1TWPD is approximated by
-
2.4 Velocity-Mismatch Bandwidth Limitation 37
0
0.5
1
1.5
2
0 0.5 1 1.5 2
Nor
mal
ized
Ban
dwid
th,
2πB
vm
/Γα
v e
Velocity Mismatch, ve/v
o
γ = 1
γ = 0
Fig. 2.19 Normalized bandwidth of γ = 1 and γ = 0 TWPDs versus
velocity mismatch. Thebroken line shows a constant approximation
(1/1.5) to the γ = 1 curve.
Bvm1 ≈
Γαve3π
(2.30)
with less than 6% error over the entire range of velocities from
completelymismatched to beyond matched,
0 ≤ ve vo ≤ 1.47. Note that (2.30) is independent of
optical velocity. Increasing the electrical velocity increases
the velocity-mismatch
bandwidth limitation, but velocity-matching is surprisingly of
almost no direct value.In fact, the mismatched case, where
ve vo > 1, is preferable to the matched case,
ve vo = 1.
The velocity-mismatch bandwidth limitation for γ = 1 in (2.30)
can be cast in amore familiar form by using (2.14) and (2.15),
Bvm1 ≈1
2πZ0C⋅ Γα1.5
. (2.31)
-
38 2. Theory of Distributed Photodetection
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
Fra
ctio
nal P
hoto
curr
ent,
|i(ω )
/Q|
Normalized Frequency, ω/Γ αve
γ = 1
Avm1 = w
1.5Γα
ve vo =1
ve vo = 0.33
Fig. 2.20 Velocity mismatch frequency responses for γ = 1 TWPDs.
Effective area approximation(broken line) is compared with velocity
matched (
v
ev
o= 1) and velocity mismatched (
v
ev
o= 0.33)
responses.
Evidently, the γ = 1 TWPD velocity-mismatch bandwidth limitation
is comparable tothe RC bandwidth limitation for a WGPD of fixed
area,
Avm1 = w
1.5Γα
. (2.32)
In fact, this area can be used to approximate the frequency
response up to the
velocity-mismatch bandwidth limitation with less than 4% error,
as illustrated in Fig.
2.20.
The effective area associated with the velocity-mismatch
bandwidth limitation for
γ = 1 TWPDs is independent of length. The effective length, lvm1
= 1.5 Γα , wouldallow only 78% internal quantum efficiency in a
WGPD of the same bandwidth.
However, the TWPD can be made physically much longer to achieve
close to 100%
internal quantum efficiency without sacrificing bandwidth.
The velocity-mismatch effective area for the γ = 0 TWPD,
-
2.4 Velocity-Mismatch Bandwidth Limitation 39
0
100
200
300
400
0 50 100 150 200 250
3 dB
Ban
dwid
th L
imita
tion
(GH
z)
i-Layer Thickness (nm)
CarrierDrift
γ = 0TWPDB
vm0
γ = 1TWPDB
vm1
WGPDB
RC
Fig. 2.21 Primary bandwidth limitation of 1 µm widt,
parallel-plate, GaAs/AlGaAs p-i-n TWPDsand WGPDs. Carrier drift,
velocity mismatch, and RC limitations are shown. 100% and
95%internal quantum efficiencies are assumed for the TWPDs and
WGPDs. The load impedance is 50 Ω.
Avm0 = w
1− ve voΓα
. (2.33)
gives the exact velocity-mismatch frequency response (2.28).
This area depends
strongly on velocity mismatch, and it goes to zero when the
velocities are matched,
corresponding to infinite bandwidth as shown in Fig. 2.18.
Fig. 2.21 shows the velocity mismatch bandwidth limitations for
γ = 0 and γ = 1TWPDs, the RC bandwidth limitation for a WGPD with
95% internal quantum
efficiency, and the carrier drift bandwidth limitation versus
i-layer thickness [19]. The
internal efficiency of both WGPDs and TWPDs is reduced by
optical scattering, free
carrier loss, absorption outside the collection field,
recombination in the active region,
and electrical losses. These effects are deemed small and
presently ignored. Fig. 2.21
suggests that the TWPDs have larger overall bandwidths than the
WGPD. Since the
γ = 1 TWPD has 100% internal quantum efficiency, it also has a
larger bandwidth-efficiency product than the WGPD.
-
40 2. Theory of Distributed Photodetection
In practice, TWPD overall bandwidth is found by replacing the
device physical
junction area in standard lumped-element calculations with the
velocity-mismatch
effective area in (2.32) or (2.33). This procedure is described
in Chapter 3.
2 . 5 Space-Charge Field Screening
A performance metric closely related to the bandwidth-efficiency
product is the
range of photodetector response linearity. Both parameters are
primary factors in
system signal-to-noise performance. Field-screening is a
fundamental mechanism that
limits the range of photodetector linearity at high
illumination. A simple model
illustrates physical phenomena that mitigate field-screening
effects in TWPDs.
Field screening arises when the dipole due to spatial separation
of photogenerated
charges significantly reduces the drift field, as illustrated
conceptually in Fig. 2.22.
The electric field in the depletion region (i-layer) is the sum
of the built-in field, the
field due to photogenerated free charge, and the fields of waves
originating elsewhereand propagating on the structure,
E = Eb + Ef + Ew . The free charge field is found by
Gauss's law and is proportional to the charge area density, Ef =
ρ f ε dx∫ ∝ σ . For
simplicity and clarity, the built-in field and the charge
densities of the electrons and
holes are approximated by rectangle functions in Fig. 2.22. The
RC time constant is
assumed zero, which is equivalent to ignoring the fields due to
the propagating waves
in a TWPD. This is a reasonable assumption for photodetectors
with transit-time
limited small-signal response.
Fig. 2.22 shows three sets of conceptual graphs at successive
times after
photogeneration of electrons and holes in the depletion region
by a short optical
pulse. The upper plots show the photogenerated charge density,
and the lower plots
show the net electric field in the depletion region. Electron
and hole velocities are
assumed equal. The field resulting from the separation of the
free charges opposes the
built-in field, and if the charge density is large enough, may
actually cancel it, as
shown in the third set of graphs. The carriers in the low-field
region then travel
slower, causing a slow component in the device photocurrent
response. Field-
screening is said to occur when the device response is
perceptibly affected.
-
2.5 Space-Charge Field Screening 41
x
electrons
holes
E
x
x
e
hx
e
h
x
E
x
E
t = 0+ t = t1 t = t2ρ f ρ f ρ f
Eb
di di di
Fig. 2.22 Conceptual illustration of field-screening mechanism.
Simplified electron and holedistributions and net electric fields
in the depletion region are plotted at three times
afterphotogeneration by a short optical pulse.
Charge build-up at the depletion region edges is ignored in Fig.
2.22, although it
may be significant in RC-limited devices. Charge reaching the
depletion region edges
contributes to the circuit current on a time scale much shorter
than the transit time in a
transit-time limited photodetector, as assumed in Fig. 2.22. A
device can always be
made transit-time limited by lowering the load impedance.
However, charge lingers at
the depletion region edges in the RC-limited case, adding to the
screening field. Thus,
field-screening effects are more pronounced in an RC-limited
device/circuit.
Local field screening occurs when the photogenerated charge area
density at apoint in the plane of the junction exceeds a critical
value,
σ y,z( ) > σ fs. The total
charge, Q, at the field-screening threshold for a thin,
uniformly illuminated, vertically
illuminated photodetector (VPD) is then
QfsV = σ fsA. (2.34)
To first order, the photogenerated charge density profile in an
in-plane illuminated
photodetector, such as a TWPD or WGPD, decays exponentially
according to
-
42 2. Theory of Distributed Photodetection
Cha
rge
Are
a D
ensi
ty, σ
Distance, z
l
σfs
00 1/Γ α
bσfs
t = 0+
Fig. 2.23 First-order photogenerated charge density profile in
an in-plane illuminated photodetectorat its field-screening
threshold. The dotted line indicates the local field-screening
threshold density.
σ z( ) = σ 0( )e−Γαzu z( ) (2.35)
as illustrated in Fig. 2.23. This nonuniform distribution gives
rise to several
phenomena that impact field-screening. These phenomena are
accounted for by
defining the field-screening threshold as the total charge when
the density at the input
according to (2.35) is above the threshold density by a factor
b. For a TWPD, this is
expressed in terms of the velocity-mismatch effective area
QfsT =
b
1.5σ fsAvm1. (2.36)
Fig. 2.24 outlines effects that impact b. The dashed line is the
simple exponential
as in Fig. 2.23. Nonlinear absorption, in which the absorption
decreases with
increasing volume density of electron-hole pairs, is depicted by
the solid line. This
effect reduces field screening, although it is compensated some
by carrier heating in
the drift field [21]. Charges in the unscreened regions are not
affected by field
-
2.5 Space-Charge Field Screening 43
Cha
rge
Are
a D
ensi
ty, σ
Distance, zl
Field-ScreenedRegion
σfs
t = 0+Nonlinear Absorption
& Carrier Heating
Longitudinal Drift& Diffusion
00 1/Γ α
UnscreenedRegion
Fig. 2.24 Photogenerated charge density profile in an in-plane
illuminated photodetector showingfield-screened and unscreened
regions, and illustrating effects of longitudinal drift and
diffusion,nonlinear absorption, and carrier heating. The
first-order expon